In Primary Mathematical Theory it is important to find the largest positive integer that divides two or more numbers without a reminder. For example, it is useful to keep the lower part fraction low. As an example, to reduce 203/377 to a lower term, you need to know that 29 is the largest positive integer separating 203 and 377. Next, 203/377 = (7) (29) / (13) (29) = 7/13. How do you find that 29 is the largest integer separating 203 and 377? One way is to determine the prime factorization and comparison factors of the two numbers. That is, we need to know 203 = (7) (29) and 377 = (13) (29). A much more efficient way is the Euclidean algorithm. The largest positive integer that divides two or more numbers without a reminder is called two or more numbers of GFM (GREATEST COMMON FACTOR). The first way to find a GCF is to find the prime factor of the number. The second method based on the Euclidean algorithm is more efficient and will be discussed here. Its main meaning is that it does not require factoring. GCF is also called the greatest common divisor, GCD. This is sometimes called the HCF I method based on the Euclidean algorithm for finding two numbers of GCFs, also called Highest Common Factor.
Step 1: Divide the big number (Dividend) by a small number (Divisor) to get some surplus.
Step 2: Divide Divisor (to be a dividend) by the remaining people (become Divisor) and get a new Remainder.
Step 3: Continue the process of sequentially splitting the Divisors that the reminder got, until you get a reminder.
Step 4: The last divisor is the GCF of the given two numbers. All these steps are displayed in one place as a single unit similar to a long division. This method will be clarified by the following example.
Example I (1): Finding the GCF of numbers 16 and 30 Solution:
16 (30) 1
16
------
14 (16) 1
14 ------
GCF 2 (14) 7
14
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0
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See the biggest common factor finding the process presentation above.
Step 1: Divide the large number (Dividend, 30) by a small number (Divisor, 16) and get Remainder 14 (the quotient is 1).
Step 2: Next, we divide Divisor (16, which will be a dividend) by the remnant (14, Divisor) and get a new Remainder 2 (quotient is 1).
Step 3: Continue the process of sequentially splitting Divinors until you get a Remainder. Split the divisor (14, dividend) by the remainder (2, divisor) and get a new remainder 0 (quotient is 7). Step 4: The last divisor, 2 is the given two 16 and 30 GCFs. Therefore, 16 and 30 GCF = 2.
Example I (2): Find the GCF of numbers 45 and 120. solution:
45 (120) 2
90
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30 (45) 1
30
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GCF 15 (30) 2
30
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0
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See the presentation of the above GCF detection process. Divide 120 by 45 and get 30 as a reminder (quotient is 2). At the next stage, 30 is the divisor and 45 is the division. This split wave 15 is a reminder (quotient is 1). In the next stage, 15 is a divisor and 30 is division. This split wave 0 is a reminder (quotient is 2). The last Divisor 15 is the GCF of the two numbers given. Therefore GCF is 45 and 120 = 15.
Example I (3): Search GCF for numbers 1066 and 46189. solution:
1066 (46189) 43
45838
------
351 (1066) 3
1053
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GCF 13 (351) 27
351
-------
0
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See the presentation of the above GCF detection process. Divide 46189 by 1066 to obtain 351 as a reminder (the quotient of 43). At the next stage, 351 is the divisor and 1066 is the division. This split wave 13 is a reminder (quotient is 3). At the next stage, 13 is a divisor and 351 is division. This split wave 0 is a reminder (quotient 27). The last divisor 13 is the GCF of the given two numbers. Therefore, GCF of 1066 and 46189 = 13. This partitioning method that finds the largest common factor is particularly useful for finding numerous GCFs. Suppose you run this example 3 with Prime Factorisation. You will understand the advantages of the superior factorization of this subcommittee.
II How to find two or more numbers of GCFs: To find more than two GCFs, first find those two GCFs. Then find the third number of GCFs and the first two numbers of GCFs so obtained. Continue this method in turn until all the numbers are over. Let's see some examples.
Example II (1): Find the GCF of numbers 60, 90, 150. Answer: First, find GCFs with numbers 60 and 90.
60 (90) 1
60
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GCF 30 (60) 2
60
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0
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Therefore, 60 and 90 GCF = 30, so that the GCF is 30 and 150. 150 is five times 30. Therefore, the GCF is 30 and 150 = 30. One of the two factors is the factor that element is the GCF of the two numbers. Therefore, GCF of numbers 60, 90, and 150 = 30.
Answer: First, let's look at the GCF of 70 and 210. It turns out that 210 is 3 times 70. Therefore, GCF is 70 and 210 = 70. Let's see if GCF is 70 and 315.
70 (315) 4
280
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GCF 35 (70) 2
70
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0
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Therefore, GCF = 35 at 70 and 315. Therefore, the GCF of 70, 210, 315 is 35 = 35.
Example II (3): Find the GCF of 1197, 5320, 4389. Solution: First find the GCF of 1197, 5320.
1197 (5320)
4788 ------
532 (1197) 2
1064
------
GCF 133 (532) 4
532
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0
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Therefore, GCF of numbers 1197 and 5320 = 133. Next, find the GCF of 133 and 4389.
GCF 133 (4389)
4389
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0
------
Therefore, GCF of 133 and 4389 = 133. Therefore, GCF of 1197, 5320, 4389 = 133.
Example II (4): Find the GCF numbers 1701, 2106, 27454. Solution method: First, find the GCF numbers 1701, 2106.
1701 (2106) 1
1701
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405 (1701) 4
1620
------
GCF 81 (405)
405
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0
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Therefore, GCF with numbers 1701, 2106 = 81. Next, let's look at the GCF of 81 and 2754.
GCF 81 (2754) 34
2754
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0
------
Therefore, GCF of 81 and 2754 = 81. GOF of SO, 1701, 2106, 2754 = 81.
